第 23721 題：Legendre符號

原題與已知條件

odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={"prime":79}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — odd prime上的zero/residue/nonresidue三值完整分類。操作=legendre-classification-table；資料={"prime":79}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-legendre-symbol

逐步推導

source operation=legendre-classification-table;original snapshot and mathematics retained

explicit ordered source output fields=[prime, class_symbol_rows, complete_partition, certificate]

Legendre ORIGINAL odd prime:ORIGINAL modulus=79;ALL possible trial divisors through square root=[2, 3, 4, 5, 6, 7, 8]

ALL [divisor,remainder] rows=[[2, 1], [3, 1], [4, 3], [5, 4], [6, 1], [7, 2], [8, 7]];prime=是

Legendre ORIGINAL nonzero root1 exact square := mul(1, 1)

Legendre ORIGINAL nonzero root1 exact square = 1

ORIGINAL root x=1 in1..78;exact square=1

square=0*79+1;canonical square class=1;ALL root indices retained

Legendre ORIGINAL nonzero root2 exact square := mul(2, 2)

Legendre ORIGINAL nonzero root2 exact square = 4

ORIGINAL root x=2 in1..78;exact square=4

square=0*79+4;canonical square class=4;ALL root indices retained

Legendre ORIGINAL nonzero root3 exact square := mul(3, 3)

Legendre ORIGINAL nonzero root3 exact square = 9

ORIGINAL root x=3 in1..78;exact square=9

square=0*79+9;canonical square class=9;ALL root indices retained

Legendre ORIGINAL nonzero root4 exact square := mul(4, 4)

Legendre ORIGINAL nonzero root4 exact square = 16

ORIGINAL root x=4 in1..78;exact square=16

square=0*79+16;canonical square class=16;ALL root indices retained

Legendre ORIGINAL nonzero root5 exact square := mul(5, 5)

Legendre ORIGINAL nonzero root5 exact square = 25

ORIGINAL root x=5 in1..78;exact square=25

square=0*79+25;canonical square class=25;ALL root indices retained

Legendre ORIGINAL nonzero root6 exact square := mul(6, 6)

Legendre ORIGINAL nonzero root6 exact square = 36

ORIGINAL root x=6 in1..78;exact square=36

square=0*79+36;canonical square class=36;ALL root indices retained

Legendre ORIGINAL nonzero root7 exact square := mul(7, 7)

Legendre ORIGINAL nonzero root7 exact square = 49

ORIGINAL root x=7 in1..78;exact square=49

square=0*79+49;canonical square class=49;ALL root indices retained

Legendre ORIGINAL nonzero root8 exact square := mul(8, 8)

Legendre ORIGINAL nonzero root8 exact square = 64

ORIGINAL root x=8 in1..78;exact square=64

square=0*79+64;canonical square class=64;ALL root indices retained

Legendre ORIGINAL nonzero root9 exact square := mul(9, 9)

Legendre ORIGINAL nonzero root9 exact square = 81

ORIGINAL root x=9 in1..78;exact square=81

square=1*79+2;canonical square class=2;ALL root indices retained

Legendre ORIGINAL nonzero root10 exact square := mul(10, 10)

Legendre ORIGINAL nonzero root10 exact square = 100

ORIGINAL root x=10 in1..78;exact square=100

square=1*79+21;canonical square class=21;ALL root indices retained

Legendre ORIGINAL nonzero root11 exact square := mul(11, 11)

Legendre ORIGINAL nonzero root11 exact square = 121

ORIGINAL root x=11 in1..78;exact square=121

square=1*79+42;canonical square class=42;ALL root indices retained

Legendre ORIGINAL nonzero root12 exact square := mul(12, 12)

Legendre ORIGINAL nonzero root12 exact square = 144

ORIGINAL root x=12 in1..78;exact square=144

square=1*79+65;canonical square class=65;ALL root indices retained

Legendre ORIGINAL nonzero root13 exact square := mul(13, 13)

Legendre ORIGINAL nonzero root13 exact square = 169

ORIGINAL root x=13 in1..78;exact square=169

square=2*79+11;canonical square class=11;ALL root indices retained

Legendre ORIGINAL nonzero root14 exact square := mul(14, 14)

Legendre ORIGINAL nonzero root14 exact square = 196

ORIGINAL root x=14 in1..78;exact square=196

square=2*79+38;canonical square class=38;ALL root indices retained

Legendre ORIGINAL nonzero root15 exact square := mul(15, 15)

Legendre ORIGINAL nonzero root15 exact square = 225

ORIGINAL root x=15 in1..78;exact square=225

square=2*79+67;canonical square class=67;ALL root indices retained

Legendre ORIGINAL nonzero root16 exact square := mul(16, 16)

Legendre ORIGINAL nonzero root16 exact square = 256

ORIGINAL root x=16 in1..78;exact square=256

square=3*79+19;canonical square class=19;ALL root indices retained

Legendre ORIGINAL nonzero root17 exact square := mul(17, 17)

Legendre ORIGINAL nonzero root17 exact square = 289

ORIGINAL root x=17 in1..78;exact square=289

square=3*79+52;canonical square class=52;ALL root indices retained

Legendre ORIGINAL nonzero root18 exact square := mul(18, 18)

Legendre ORIGINAL nonzero root18 exact square = 324

ORIGINAL root x=18 in1..78;exact square=324

square=4*79+8;canonical square class=8;ALL root indices retained

Legendre ORIGINAL nonzero root19 exact square := mul(19, 19)

Legendre ORIGINAL nonzero root19 exact square = 361

ORIGINAL root x=19 in1..78;exact square=361

square=4*79+45;canonical square class=45;ALL root indices retained

Legendre ORIGINAL nonzero root20 exact square := mul(20, 20)

Legendre ORIGINAL nonzero root20 exact square = 400

ORIGINAL root x=20 in1..78;exact square=400

square=5*79+5;canonical square class=5;ALL root indices retained

Legendre ORIGINAL nonzero root21 exact square := mul(21, 21)

Legendre ORIGINAL nonzero root21 exact square = 441

ORIGINAL root x=21 in1..78;exact square=441

square=5*79+46;canonical square class=46;ALL root indices retained

Legendre ORIGINAL nonzero root22 exact square := mul(22, 22)

Legendre ORIGINAL nonzero root22 exact square = 484

ORIGINAL root x=22 in1..78;exact square=484

square=6*79+10;canonical square class=10;ALL root indices retained

Legendre ORIGINAL nonzero root23 exact square := mul(23, 23)

Legendre ORIGINAL nonzero root23 exact square = 529

ORIGINAL root x=23 in1..78;exact square=529

square=6*79+55;canonical square class=55;ALL root indices retained

Legendre ORIGINAL nonzero root24 exact square := mul(24, 24)

Legendre ORIGINAL nonzero root24 exact square = 576

ORIGINAL root x=24 in1..78;exact square=576

square=7*79+23;canonical square class=23;ALL root indices retained

Legendre ORIGINAL nonzero root25 exact square := mul(25, 25)

Legendre ORIGINAL nonzero root25 exact square = 625

ORIGINAL root x=25 in1..78;exact square=625

square=7*79+72;canonical square class=72;ALL root indices retained

Legendre ORIGINAL nonzero root26 exact square := mul(26, 26)

Legendre ORIGINAL nonzero root26 exact square = 676

ORIGINAL root x=26 in1..78;exact square=676

square=8*79+44;canonical square class=44;ALL root indices retained

Legendre ORIGINAL nonzero root27 exact square := mul(27, 27)

Legendre ORIGINAL nonzero root27 exact square = 729

ORIGINAL root x=27 in1..78;exact square=729

square=9*79+18;canonical square class=18;ALL root indices retained

Legendre ORIGINAL nonzero root28 exact square := mul(28, 28)

Legendre ORIGINAL nonzero root28 exact square = 784

ORIGINAL root x=28 in1..78;exact square=784

square=9*79+73;canonical square class=73;ALL root indices retained

Legendre ORIGINAL nonzero root29 exact square := mul(29, 29)

Legendre ORIGINAL nonzero root29 exact square = 841

ORIGINAL root x=29 in1..78;exact square=841

square=10*79+51;canonical square class=51;ALL root indices retained

Legendre ORIGINAL nonzero root30 exact square := mul(30, 30)

Legendre ORIGINAL nonzero root30 exact square = 900

ORIGINAL root x=30 in1..78;exact square=900

square=11*79+31;canonical square class=31;ALL root indices retained

Legendre ORIGINAL nonzero root31 exact square := mul(31, 31)

Legendre ORIGINAL nonzero root31 exact square = 961

ORIGINAL root x=31 in1..78;exact square=961

square=12*79+13;canonical square class=13;ALL root indices retained

Legendre ORIGINAL nonzero root32 exact square := mul(32, 32)

Legendre ORIGINAL nonzero root32 exact square = 1024

ORIGINAL root x=32 in1..78;exact square=1024

square=12*79+76;canonical square class=76;ALL root indices retained

Legendre ORIGINAL nonzero root33 exact square := mul(33, 33)

Legendre ORIGINAL nonzero root33 exact square = 1089

ORIGINAL root x=33 in1..78;exact square=1089

square=13*79+62;canonical square class=62;ALL root indices retained

Legendre ORIGINAL nonzero root34 exact square := mul(34, 34)

Legendre ORIGINAL nonzero root34 exact square = 1156

ORIGINAL root x=34 in1..78;exact square=1156

square=14*79+50;canonical square class=50;ALL root indices retained

Legendre ORIGINAL nonzero root35 exact square := mul(35, 35)

Legendre ORIGINAL nonzero root35 exact square = 1225

ORIGINAL root x=35 in1..78;exact square=1225

square=15*79+40;canonical square class=40;ALL root indices retained

Legendre ORIGINAL nonzero root36 exact square := mul(36, 36)

Legendre ORIGINAL nonzero root36 exact square = 1296

ORIGINAL root x=36 in1..78;exact square=1296

square=16*79+32;canonical square class=32;ALL root indices retained

Legendre ORIGINAL nonzero root37 exact square := mul(37, 37)

Legendre ORIGINAL nonzero root37 exact square = 1369

ORIGINAL root x=37 in1..78;exact square=1369

square=17*79+26;canonical square class=26;ALL root indices retained

Legendre ORIGINAL nonzero root38 exact square := mul(38, 38)

Legendre ORIGINAL nonzero root38 exact square = 1444

ORIGINAL root x=38 in1..78;exact square=1444

square=18*79+22;canonical square class=22;ALL root indices retained

Legendre ORIGINAL nonzero root39 exact square := mul(39, 39)

Legendre ORIGINAL nonzero root39 exact square = 1521

ORIGINAL root x=39 in1..78;exact square=1521

square=19*79+20;canonical square class=20;ALL root indices retained

Legendre ORIGINAL nonzero root40 exact square := mul(40, 40)

Legendre ORIGINAL nonzero root40 exact square = 1600

ORIGINAL root x=40 in1..78;exact square=1600

square=20*79+20;canonical square class=20;ALL root indices retained

Legendre ORIGINAL nonzero root41 exact square := mul(41, 41)

Legendre ORIGINAL nonzero root41 exact square = 1681

ORIGINAL root x=41 in1..78;exact square=1681

square=21*79+22;canonical square class=22;ALL root indices retained

Legendre ORIGINAL nonzero root42 exact square := mul(42, 42)

Legendre ORIGINAL nonzero root42 exact square = 1764

ORIGINAL root x=42 in1..78;exact square=1764

square=22*79+26;canonical square class=26;ALL root indices retained

Legendre ORIGINAL nonzero root43 exact square := mul(43, 43)

Legendre ORIGINAL nonzero root43 exact square = 1849

ORIGINAL root x=43 in1..78;exact square=1849

square=23*79+32;canonical square class=32;ALL root indices retained

Legendre ORIGINAL nonzero root44 exact square := mul(44, 44)

Legendre ORIGINAL nonzero root44 exact square = 1936

ORIGINAL root x=44 in1..78;exact square=1936

square=24*79+40;canonical square class=40;ALL root indices retained

Legendre ORIGINAL nonzero root45 exact square := mul(45, 45)

Legendre ORIGINAL nonzero root45 exact square = 2025

ORIGINAL root x=45 in1..78;exact square=2025

square=25*79+50;canonical square class=50;ALL root indices retained

Legendre ORIGINAL nonzero root46 exact square := mul(46, 46)

Legendre ORIGINAL nonzero root46 exact square = 2116

ORIGINAL root x=46 in1..78;exact square=2116

square=26*79+62;canonical square class=62;ALL root indices retained

Legendre ORIGINAL nonzero root47 exact square := mul(47, 47)

Legendre ORIGINAL nonzero root47 exact square = 2209

ORIGINAL root x=47 in1..78;exact square=2209

square=27*79+76;canonical square class=76;ALL root indices retained

Legendre ORIGINAL nonzero root48 exact square := mul(48, 48)

Legendre ORIGINAL nonzero root48 exact square = 2304

ORIGINAL root x=48 in1..78;exact square=2304

square=29*79+13;canonical square class=13;ALL root indices retained

Legendre ORIGINAL nonzero root49 exact square := mul(49, 49)

Legendre ORIGINAL nonzero root49 exact square = 2401

ORIGINAL root x=49 in1..78;exact square=2401

square=30*79+31;canonical square class=31;ALL root indices retained

Legendre ORIGINAL nonzero root50 exact square := mul(50, 50)

Legendre ORIGINAL nonzero root50 exact square = 2500

ORIGINAL root x=50 in1..78;exact square=2500

square=31*79+51;canonical square class=51;ALL root indices retained

Legendre ORIGINAL nonzero root51 exact square := mul(51, 51)

Legendre ORIGINAL nonzero root51 exact square = 2601

ORIGINAL root x=51 in1..78;exact square=2601

square=32*79+73;canonical square class=73;ALL root indices retained

Legendre ORIGINAL nonzero root52 exact square := mul(52, 52)

Legendre ORIGINAL nonzero root52 exact square = 2704

ORIGINAL root x=52 in1..78;exact square=2704

square=34*79+18;canonical square class=18;ALL root indices retained

Legendre ORIGINAL nonzero root53 exact square := mul(53, 53)

Legendre ORIGINAL nonzero root53 exact square = 2809

ORIGINAL root x=53 in1..78;exact square=2809

square=35*79+44;canonical square class=44;ALL root indices retained

Legendre ORIGINAL nonzero root54 exact square := mul(54, 54)

Legendre ORIGINAL nonzero root54 exact square = 2916

ORIGINAL root x=54 in1..78;exact square=2916

square=36*79+72;canonical square class=72;ALL root indices retained

Legendre ORIGINAL nonzero root55 exact square := mul(55, 55)

Legendre ORIGINAL nonzero root55 exact square = 3025

ORIGINAL root x=55 in1..78;exact square=3025

square=38*79+23;canonical square class=23;ALL root indices retained

Legendre ORIGINAL nonzero root56 exact square := mul(56, 56)

Legendre ORIGINAL nonzero root56 exact square = 3136

ORIGINAL root x=56 in1..78;exact square=3136

square=39*79+55;canonical square class=55;ALL root indices retained

Legendre ORIGINAL nonzero root57 exact square := mul(57, 57)

Legendre ORIGINAL nonzero root57 exact square = 3249

ORIGINAL root x=57 in1..78;exact square=3249

square=41*79+10;canonical square class=10;ALL root indices retained

Legendre ORIGINAL nonzero root58 exact square := mul(58, 58)

Legendre ORIGINAL nonzero root58 exact square = 3364

ORIGINAL root x=58 in1..78;exact square=3364

square=42*79+46;canonical square class=46;ALL root indices retained

Legendre ORIGINAL nonzero root59 exact square := mul(59, 59)

Legendre ORIGINAL nonzero root59 exact square = 3481

ORIGINAL root x=59 in1..78;exact square=3481

square=44*79+5;canonical square class=5;ALL root indices retained

Legendre ORIGINAL nonzero root60 exact square := mul(60, 60)

Legendre ORIGINAL nonzero root60 exact square = 3600

ORIGINAL root x=60 in1..78;exact square=3600

square=45*79+45;canonical square class=45;ALL root indices retained

Legendre ORIGINAL nonzero root61 exact square := mul(61, 61)

Legendre ORIGINAL nonzero root61 exact square = 3721

ORIGINAL root x=61 in1..78;exact square=3721

square=47*79+8;canonical square class=8;ALL root indices retained

Legendre ORIGINAL nonzero root62 exact square := mul(62, 62)

Legendre ORIGINAL nonzero root62 exact square = 3844

ORIGINAL root x=62 in1..78;exact square=3844

square=48*79+52;canonical square class=52;ALL root indices retained

Legendre ORIGINAL nonzero root63 exact square := mul(63, 63)

Legendre ORIGINAL nonzero root63 exact square = 3969

ORIGINAL root x=63 in1..78;exact square=3969

square=50*79+19;canonical square class=19;ALL root indices retained

Legendre ORIGINAL nonzero root64 exact square := mul(64, 64)

Legendre ORIGINAL nonzero root64 exact square = 4096

ORIGINAL root x=64 in1..78;exact square=4096

square=51*79+67;canonical square class=67;ALL root indices retained

Legendre ORIGINAL nonzero root65 exact square := mul(65, 65)

Legendre ORIGINAL nonzero root65 exact square = 4225

ORIGINAL root x=65 in1..78;exact square=4225

square=53*79+38;canonical square class=38;ALL root indices retained

Legendre ORIGINAL nonzero root66 exact square := mul(66, 66)

Legendre ORIGINAL nonzero root66 exact square = 4356

ORIGINAL root x=66 in1..78;exact square=4356

square=55*79+11;canonical square class=11;ALL root indices retained

Legendre ORIGINAL nonzero root67 exact square := mul(67, 67)

Legendre ORIGINAL nonzero root67 exact square = 4489

ORIGINAL root x=67 in1..78;exact square=4489

square=56*79+65;canonical square class=65;ALL root indices retained

Legendre ORIGINAL nonzero root68 exact square := mul(68, 68)

Legendre ORIGINAL nonzero root68 exact square = 4624

ORIGINAL root x=68 in1..78;exact square=4624

square=58*79+42;canonical square class=42;ALL root indices retained

Legendre ORIGINAL nonzero root69 exact square := mul(69, 69)

Legendre ORIGINAL nonzero root69 exact square = 4761

ORIGINAL root x=69 in1..78;exact square=4761

square=60*79+21;canonical square class=21;ALL root indices retained

Legendre ORIGINAL nonzero root70 exact square := mul(70, 70)

Legendre ORIGINAL nonzero root70 exact square = 4900

ORIGINAL root x=70 in1..78;exact square=4900

square=62*79+2;canonical square class=2;ALL root indices retained

Legendre ORIGINAL nonzero root71 exact square := mul(71, 71)

Legendre ORIGINAL nonzero root71 exact square = 5041

ORIGINAL root x=71 in1..78;exact square=5041

square=63*79+64;canonical square class=64;ALL root indices retained

Legendre ORIGINAL nonzero root72 exact square := mul(72, 72)

Legendre ORIGINAL nonzero root72 exact square = 5184

ORIGINAL root x=72 in1..78;exact square=5184

square=65*79+49;canonical square class=49;ALL root indices retained

Legendre ORIGINAL nonzero root73 exact square := mul(73, 73)

Legendre ORIGINAL nonzero root73 exact square = 5329

ORIGINAL root x=73 in1..78;exact square=5329

square=67*79+36;canonical square class=36;ALL root indices retained

Legendre ORIGINAL nonzero root74 exact square := mul(74, 74)

Legendre ORIGINAL nonzero root74 exact square = 5476

ORIGINAL root x=74 in1..78;exact square=5476

square=69*79+25;canonical square class=25;ALL root indices retained

Legendre ORIGINAL nonzero root75 exact square := mul(75, 75)

Legendre ORIGINAL nonzero root75 exact square = 5625

ORIGINAL root x=75 in1..78;exact square=5625

square=71*79+16;canonical square class=16;ALL root indices retained

Legendre ORIGINAL nonzero root76 exact square := mul(76, 76)

Legendre ORIGINAL nonzero root76 exact square = 5776

ORIGINAL root x=76 in1..78;exact square=5776

square=73*79+9;canonical square class=9;ALL root indices retained

Legendre ORIGINAL nonzero root77 exact square := mul(77, 77)

Legendre ORIGINAL nonzero root77 exact square = 5929

ORIGINAL root x=77 in1..78;exact square=5929

square=75*79+4;canonical square class=4;ALL root indices retained

Legendre ORIGINAL nonzero root78 exact square := mul(78, 78)

Legendre ORIGINAL nonzero root78 exact square = 6084

ORIGINAL root x=78 in1..78;exact square=6084

square=77*79+1;canonical square class=1;ALL root indices retained

ORIGINAL prime=79;COMPLETE nonzero square table=[[1, 1], [2, 4], [3, 9], [4, 16], [5, 25], [6, 36], [7, 49], [8, 64], [9, 2], [10, 21], [11, 42], [12, 65], [13, 11], [14, 38], [15, 67], [16, 19], [17, 52], [18, 8], [19, 45], [20, 5], [21, 46], [22, 10], [23, 55], [24, 23], [25, 72], [26, 44], [27, 18], [28, 73], [29, 51], [30, 31], [31, 13], [32, 76], [33, 62], [34, 50], [35, 40], [36, 32], [37, 26], [38, 22], [39, 20], [40, 20], [41, 22], [42, 26], [43, 32], [44, 40], [45, 50], [46, 62], [47, 76], [48, 13], [49, 31], [50, 51], [51, 73], [52, 18], [53, 44], [54, 72], [55, 23], [56, 55], [57, 10], [58, 46], [59, 5], [60, 45], [61, 8], [62, 52], [63, 19], [64, 67], [65, 38], [66, 11], [67, 65], [68, 42], [69, 21], [70, 2], [71, 64], [72, 49], [73, 36], [74, 25], [75, 16], [76, 9], [77, 4], [78, 1]];nonzero square classes=[1, 2, 4, 5, 8, 9, 10, 11, 13, 16, 18, 19, 20, 21, 22, 23, 25, 26, 31, 32, 36, 38, 40, 42, 44, 45, 46, 49, 50, 51, 52, 55, 62, 64, 65, 67, 72, 73, 76]

for EACH normalized class r,first r0 ->symbol0;else square-table member ->symbol1;otherwise ->symbol-1;these cases partition ALL0..p-1

逐步反向核對

• S159 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S158 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S157 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root78 exact square，不以小數近似替代。
• S156 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S155 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root77 exact square，不以小數近似替代。
• S154 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S153 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root76 exact square，不以小數近似替代。
• S152 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S151 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root75 exact square，不以小數近似替代。
• S150 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S149 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root74 exact square，不以小數近似替代。
• S148 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S147 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root73 exact square，不以小數近似替代。
• S146 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S145 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root72 exact square，不以小數近似替代。
• S144 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S143 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root71 exact square，不以小數近似替代。
• S142 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S141 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root70 exact square，不以小數近似替代。
• S140 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S139 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root69 exact square，不以小數近似替代。
• S138 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S137 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root68 exact square，不以小數近似替代。
• S136 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S135 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root67 exact square，不以小數近似替代。
• S134 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S133 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root66 exact square，不以小數近似替代。
• S132 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S131 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root65 exact square，不以小數近似替代。
• S130 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S129 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root64 exact square，不以小數近似替代。
• S128 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S127 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root63 exact square，不以小數近似替代。
• S126 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S125 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root62 exact square，不以小數近似替代。
• S124 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S123 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root61 exact square，不以小數近似替代。
• S122 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S121 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root60 exact square，不以小數近似替代。
• S120 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S119 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root59 exact square，不以小數近似替代。
• S118 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S117 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root58 exact square，不以小數近似替代。
• S116 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S115 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root57 exact square，不以小數近似替代。
• S114 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S113 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root56 exact square，不以小數近似替代。
• S112 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S111 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root55 exact square，不以小數近似替代。
• S110 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S109 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root54 exact square，不以小數近似替代。
• S108 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S107 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root53 exact square，不以小數近似替代。
• S106 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S105 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root52 exact square，不以小數近似替代。
• S104 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S103 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root51 exact square，不以小數近似替代。
• S102 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S101 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root50 exact square，不以小數近似替代。
• S100 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S099 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root49 exact square，不以小數近似替代。
• S098 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S097 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root48 exact square，不以小數近似替代。
• S096 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S095 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root47 exact square，不以小數近似替代。
• S094 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S093 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root46 exact square，不以小數近似替代。
• S092 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S091 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root45 exact square，不以小數近似替代。
• S090 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S089 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root44 exact square，不以小數近似替代。
• S088 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S087 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root43 exact square，不以小數近似替代。
• S086 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S085 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root42 exact square，不以小數近似替代。
• S084 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S083 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root41 exact square，不以小數近似替代。
• S082 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S081 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root40 exact square，不以小數近似替代。
• S080 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S079 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root39 exact square，不以小數近似替代。
• S078 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S077 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root38 exact square，不以小數近似替代。
• S076 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S075 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root37 exact square，不以小數近似替代。
• S074 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S073 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root36 exact square，不以小數近似替代。
• S072 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S071 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root35 exact square，不以小數近似替代。
• S070 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S069 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root34 exact square，不以小數近似替代。
• S068 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S067 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root33 exact square，不以小數近似替代。
• S066 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S065 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root32 exact square，不以小數近似替代。
• S064 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S063 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root31 exact square，不以小數近似替代。
• S062 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S061 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root30 exact square，不以小數近似替代。
• S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S059 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root29 exact square，不以小數近似替代。
• S058 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S057 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root28 exact square，不以小數近似替代。
• S056 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S055 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root27 exact square，不以小數近似替代。
• S054 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S053 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root26 exact square，不以小數近似替代。
• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root25 exact square，不以小數近似替代。
• S050 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S049 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root24 exact square，不以小數近似替代。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root23 exact square，不以小數近似替代。
• S046 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S045 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root22 exact square，不以小數近似替代。
• S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S043 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root21 exact square，不以小數近似替代。
• S042 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S041 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root20 exact square，不以小數近似替代。
• S040 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S039 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root19 exact square，不以小數近似替代。
• S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root18 exact square，不以小數近似替代。
• S036 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S035 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root17 exact square，不以小數近似替代。
• S034 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S033 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root16 exact square，不以小數近似替代。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root15 exact square，不以小數近似替代。
• S030 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S029 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root14 exact square，不以小數近似替代。
• S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S027 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root13 exact square，不以小數近似替代。
• S026 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S025 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root12 exact square，不以小數近似替代。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root11 exact square，不以小數近似替代。
• S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root10 exact square，不以小數近似替代。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root9 exact square，不以小數近似替代。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root8 exact square，不以小數近似替代。
• S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root7 exact square，不以小數近似替代。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root6 exact square，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root5 exact square，不以小數近似替代。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root4 exact square，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root3 exact square，不以小數近似替代。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root2 exact square，不以小數近似替代。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 以精確有理數重算 Legendre ORIGINAL nonzero root1 exact square，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(79, [[0, 0], [1, 1], [2, 1], [3, -1], [4, 1], [5, 1], [6, -1], [7, -1], [8, 1], [9, 1], [10, 1], [11, 1], [12, -1], [13, 1], [14, -1], [15, -1], [16, 1], [17, -1], [18, 1], [19, 1], [20, 1], [21, 1], [22, 1], [23, 1], [24, -1], [25, 1], [26, 1], [27, -1], [28, -1], [29, -1], [30, -1], [31, 1], [32, 1], [33, -1], [34, -1], [35, -1], [36, 1], [37, -1], [38, 1], [39, -1], [40, 1], [41, -1], [42, 1], [43, -1], [44, 1], [45, 1], [46, 1], [47, -1], [48, -1], [49, 1], [50, 1], [51, 1], [52, 1], [53, -1], [54, -1], [55, 1], [56, -1], [57, -1], [58, -1], [59, -1], [60, -1], [61, -1], [62, 1], [63, -1], [64, 1], [65, 1], [66, -1], [67, 1], [68, -1], [69, -1], [70, -1], [71, -1], [72, 1], [73, 1], [74, -1], [75, -1], [76, 1], [77, -1], [78, -1]], 是, zero_residue_nonresidue_partition_is_complete)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。